Statistical analysis of hard X-ray radiation at the PAL-XFEL facility performed by Hanbury Brown and Twiss interferometry

Statistical properties of the hard X-ray free-electron laser PAL-XFEL were studied by Hanbury Brown and Twiss interferometry. The results demonstrate high spatial coherence and short average pulse duration of this facility at 10 keV photon energy.


S1. Data processing
In this experiment, we simultaneously measured intensity information in the spectral and spatial domains. The number of pulses analysed in this work for each bunch charge and operation condition of the PAL XFEL are listed in Table S1. In order to extract accurate information of the data, dark images of about 1000 shots were obtained in each condition, the average value was calculated, and the dark image was subtracted from each data in spectral and spatial domains. After processing of these data, a projection along the vertical direction was performed in 2D spectral detector. One example is shown in Fig. S1 (a,b). In each condition, the average profile from all pulses of the one-dimensional spectra was obtained, the zero value of ΔE was set through the fitting of Gaussian distribution, and the value of full width of half maximum (FWHM) for energy distribution was obtained (see Fig. S1 (c)).
For each pulse, a spatial image was also projected vertically and horizontally to obtain the onedimensional profile. After that, the average value for all pulses was calculated, the zero position was set through the fitting of the Gaussian distribution in each direction, and the FWHM for the beam size was calculated (see Fig. S1 (e,f)).

S2. Monochromator drift corrections
During the measurements for the case of SASE with the monochromator, we observed the vertical position drifts of the monochromator (see Fig. S2). So, we set the sections where the drift was happening rapidly, calculated the linear regression for each section, and then corrected it (see Fig. S2 (b,c)). This correction was performed for all monochromatic radiation modes.    (a,b) SASE radiation, (e,f) SASE monochromatic radiation, (i,j) self-seeding regime of operation.
(g,h) Same in self-seeding linear mode of operation. All results presented in this figure correspond to the 200 pC bunch charge.
S4. An average spectral profile fit As it was described in the main text to determine coherence time one needs to obtain an averaged spectral profile. We noticed that in the case of PAL XFEL an averaged spectral profile is well fitted by two Gaussian functions as where A1 and A2 are scaling coefficients, and are the centres of each Gaussian line and σ1 and σ2 are their rms values. Results of such fitting are summarized in Table S2 where I is the integrated intensity for a single pulse, 〈 〉 is the average intensity from all pulses and M is the number of modes.
According to the FEL theory (Saldin et al., 2000), the number of modes M is inversely proportional to the normalized dispersion of the intensity distribution where is the standard deviation of the intensity distribution. The results of this analysis were summarized in Table S3. As we can see from Figs. S6-S8 Gamma distribution is observed mostly in monochromatic regime of operation when there are few modes are contributing to the total intensity. In the SASE mode number of modes is large (about 100) and analysis by Gamma distribution is not working. In self-seeding mode pulse distribution is complicated and will need a special analysis.

S6.1. Spectral analysis
The normalized spectral g (2) (ω1,ω2)-correlation function has the following form where ( ) and ( ) are the intensities of the wave field in spectral representation, is the central frequency, and averaging denoted by brackets <…> is performed over a large ensemble of different realizations of the wave field. These spectral correlation functions are presented in Figs. S9-S12 and are discussed in the main part of the paper.

S6.2. Spatial analysis
The normalized spatial second-order correlation function is expressed as Here (